Tameness and Quantum Field Theory

A paper by Michael R. Douglas, Thomas W. Grimm, and Lorenz Schlechter appeared on the archive yesterday. Admittedly, I haven’t had time yet to properly work through it. From what I quickly skimmed last night, the paper mainly concentrates on the notion of tameness and how tame classes of functions can be applied in the arena of renormalisable quantum field theories. This strikes me as interesting, if not curious, for several reasons. My primary interest is that, in their analysis of tame functions, there is potentially some contact (albeit I don’t know how deeply, if in fact at all) with some observations that we’ve been making in ongoing work on the construction of a canonical class of (what seem to be deeply stringy) regulators. I mentioned this work briefly in passing in the last post.

As I said, I need to sit down with the paper and see what is going on. But, as a short note, I wanted to highlight this concept of tameness (from a maths perspective) and why it is rather curious. As I currently understand, how we might define a tame function for example, begins with the notion of o-minimal structures.

Definition 1 (o-minimal structures) Given a Euclidean vector space ${\textbf{R}^n}$, an o-minimal structure on ${(\textbf{R}^n, +, .)}$ is a sequence of Boolean algebras ${\mathcal{O}=\{ \mathcal{O}_n \}}$ of definable subsets of ${\textbf{R}^n_{+}}$ such that for each ${n \in \textbf{N}}$

i) if ${A}$ belongs to ${\mathcal{O}_n}$, then ${A \times R}$ and ${R \times A}$ belong to ${\mathcal{O}_{n+1}}$;

ii) if ${\Pi : \textbf{R}^{n+1} \rightarrow \textbf{R}}$ is the canonical projection onto ${\textbf{R}^n}$ then for any ${A \in \mathcal{O}_{n+1}}$, the set ${\Pi (A)}$ belongs to ${\mathcal{O}_n}$;

iii) it is required ${\mathcal{O}_n}$ contains the family of algebraic subsets of ${\textbf{R}^n}$

$\displaystyle \{ x \in \textbf{R}^n : p(x) = 0 \}$

in which ${p : \textbf{R}^n \rightarrow \textbf{R}}$ is polynomial.

iv) the elements of ${\mathcal{O}_1}$ are the finite unions of intervals and points.

The concept of tameness appears, then, in relation to the definable subsets of ${\textbf{R}^n}$. These definable subsets can be considered such that a map ${F : S \subset \textbf{R}^n \rightarrow \textbf{R}^m}$ is definable in ${\mathcal{O}}$ if its graph is ${\mathcal{O}}$ definable precisely as a subset of ${\textbf{R}^n \times \textbf{R}^m}$. From this, we can define the concept of tameness as follows:

Definition 2 (tame) A subset ${A}$ of ${\textbf{R}^n}$ is described as tame if for every ${r > 0}$ there exists an o-minimal structure ${\mathcal{O}}$ over ${\textbf{R}}$ such that the intersection of ${A}$ with ${[-r,r]^n}$ is definable in ${\mathcal{O}}$.

So the o-minimal structure is a fundamental object, and it collects subsets of each ${\textbf{R}^n}$. These subsets are then definable on the structure, in this case we denote the structure ${\mathcal{O}}$. These are the sets that are said to underline the so-called tame topology inasmuch that, as it is required that such a map ${F : S \subset \textbf{R}^n \rightarrow \textbf{R}^m}$ between definable sets has a graph that is also definable, then the maps themselves are be used to define $\mathcal{O}$-definable manifolds and topological spaces. So, as far as I can tell, one is essentially playing with semi-algebraic subsets and thus the axioms of semi-algebraic geometry. The actual power of this formalism, as far as the Douglas, Grimm, and Schlechter are concerned, seems to then be in the precise nature of tame functions that enjoy strong piecewise smoothness properties, which, as it relates to their concerns of finiteness in QFT, makes some sense. In our own ongoing work, indeed much time has been spent studying ${C^{\infty}}$ functions for the reasons they outline.

As a slight aside, it is noteworthy, as they say in the opening paragraphs, that there indeed exists many real smooth functions which are not analytic; but then if ${f(x)}$ is analytic, the Taylor series at ${x = 0}$ would have a finite radius of convergence that determines the function up to some ${a > 0}$. For some compact interval, it is certainly interesting how one could break this up into a finite set of subintervals, in which there would exists a convergent series expansion. I am moving away from the purpose of this post, but this finiteness property serves as a point of motivation.

A main result of the paper is to show that Feynman amplitudes are hence tame functions (of particle masses, coupling constants, momenta, or equivalent Lorentz invariant quantities). This leads to the theorem that for any renormalisable quantum field theory with finitely many particles and interactions, all amplitudes with finitely many loops and external legs are definable in the o-minimal structure.

I also find it noteworthy that, as a future direction, the authors want to look at resurgence in QFT. I’ve been learning about this recently, particularly beginning from the idea of trans-series expansions that extends the notion of perturbative expansions to include instanton corrections. From what I currently understand, often people still use a hard cutoff; it could be that our regularisation prescription might approach resurgence differently.

All food for thought.